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Abstract—We present a subdomain formulation of the periodic method of moments (PMM) with thin-wire kernel for analyzing frequency-selective surfaces (FSSs) ...
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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 53, NO. 10, OCTOBER 2005

Numerical Convergence in Periodic Method of Moments Analysis of Frequency-Selective Surfaces Based on Wire Elements John Blackburn and Luk R. Arnaut, Senior Member, IEEE

Abstract—We present a subdomain formulation of the periodic method of moments (PMM) with thin-wire kernel for analyzing frequency-selective surfaces (FSSs) with rectilinear wire-type elements. Analysis of the convergence of the impedance matrix for a FSS with aligned unidirectional elements indicates the effect of individual oscillatory and decaying components. For the individual impedance elements of this FSS, we prove and demonstrate the universality of their envelopes as a function of shell size in the spectral wire segments, the PMM converges according to domain. For ( 4 ). The dependence on the order of polynomial basis functions shows a geometric progression. The theory is also applied to a single-layer FSS having asymmetrically split segmented rings. Index Terms—Convergence of numerical methods, frequencyselective surfaces (FSSs), modelling, method of moments.

I. INTRODUCTION

T

HE conventional method of moments (CMM) [1], [2] has been extensively used to model radiation and scattering from relatively simple structures. For complex, large, or periodic structures, efficient and accurate extensions such as the periodic method of moments (PMM) are required. Surface formulations are widely used for conductive patches deposited onto a dielectric substrate. Here, we consider a PMM with thin-wire kernel for modeling wire loop-type frequency-selective surfaces (FSSs). An example that uses asymmetrically split rings is shown in Fig. 8. The thin-wire formulation conforms better to this element shape, allows for more precise modeling of small gaps, and minimizes the associated edge effects. The rate of convergence of the moment method governs its efficiency in reaching a specified level of accuracy. Most studies to date relate to entire-domain formulations using a spectral approach. Here we use a subdomain formulation having more general applicability with regard to element geometry. Particular issues studied include absolute versus relative convergence [3]–[6] and its relation to the matrix condition number [7], [8], relevance of edge and continuity conditions for entire-domain functions [3], [5], and truncation rules [6], [9], although these results are specific to the chosen type of integral equation formulation (convolutional Green’s function, spectral Galerkin method, plane-wave expansion) and basis function. Studies of

convergence for subdomain formulations are relatively scant, e.g., [10]. Although all interactions in an infinite FSS are rigorously taken into account in the integral representation of the PMM, the array elements require discretization to convert to a calculable finite-sized matrix problem. This raises the question of the effect of matrix truncation on mutual impedances and reflectivity. This paper is to be contrasted with studies of physically finite periodic arrays, e.g., [11]–[13]. Except for the use of Poisson’s sum rule, we also do not address acceleration of convergence (e.g., using transformations [14], [15], conjugate gradient and factorization methods for iterative impedance calculations [16], [17], multilevel or fast multipole techniques [18], etc.), which constitute important but separate issues regarding improvement of numerical convergence in real time. In this paper, the convergence of the PMM for an FSS with aligned unidirectional straight-wire elements as a function of matrix size is shown to consist of distinct oscillatory and monotonic contributions. These are quantified algebraically, and their relation to FSS periodicity and operating frequency is examined. For the respective impedance elements of this FSS, the universality of their envelope as a function of spectral shell size is demonstrated. We show that the PMM computation time is . A method for efficient monitoring of overall convergence is proposed. II. PMM INTEGRAL FORMULATION Consider a single-layer planar FSS of infinite extent in and -directions with periods and [Fig. 1(a)]. The unit cell contains an arbitrarily shaped wire of length within this plane. Since currents are periodic across unit cells [19], only a reference unit cell needs calculating. The total scattered field then follows from superposition over all cells in configurational space. To accelerate convergence, Poisson’s sum rule may be utilized to transform the summation to the spectral domain, yielding the total scattered field as [20, ch. 4]

(1) Manuscript received August 6, 2004; revised March 29, 2005. This work was supported by Ofcom, U.K., under Spectrum Efficiency Scheme Contract AY4464. The authors are with the National Physical Laboratory, Teddington TW11 0LW, U.K. (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2005.856336

Here, is the free-space impedance, is the running distance is the local current in the wire, along the reference wire, is the wire trajectory, is an arbitrary observation locadefines the direction of the scattered wave (see tion in space,

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BLACKBURN et al.: NUMERICAL CONVERGENCE IN PERIODIC METHOD OF MOMENTS

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below),

is the wave polarization vector (see below), and is the propagation constant in free space. In (1), “ ” (“ ”) refers to propagation in positive (negative) direction, i.e., to the right (left) of the FSS. The depend on the inteand and on the direction of the incident wave vector gers

(2) The constitute a spectrum of propagating scattered plane waves (for real ) and evanescent surface waves (for imaginary ). The contributions of the latter are important in the calculation of matrix impedance elements. The polarization vector is , where is a local unit vector in the direction of the wire axis, pointing toward increasing . The FSS is illuminated by a plane wave with linear polarization in the -plane and with arbitrary direction of propagation [Fig. 1(a)]. Imposing the PEC tangential boundary condition on the wire surface, i.e., (3)

Fig. 1. (a) Side view of planar FSS illuminated by plane wave incident . along . The spectrum of scattered plane waves is characterized by The pair (n ; n ) = (0; 0) corresponds to forward scattering; all other pairs represent grating lobes. (b) Wire segments in a unit cell. Nodes at points (n) are connected via wire segments directed along unit vectors ^ (n ), with ^ (1+) = ^ (2 ), etc. Triangle functions are shown as dashed lines. I (n)T (l) is shown with a solid line. The overall current

s

R

p

r

p 6

p 0

(8) in which either “ ” or “ ” signs may be used because (planar FSS), and with the pattern function

results in an integral equation that may be solved for , after discretization and conversion to a matrix equation, as will be detailed next. III. PMM MATRIX FORMULATION FOR TRIANGULAR BASIS FUNCTIONS Following the CMM, we expand the unknown current using a set of basis functions with expansion coefficients dictated by (3). We use triangular basis functions

We now impose (3) for all and test using Galerkin’s method. , and integrating Substituting (6) into (3), multiplying by -matrix equation with [21] yields an

(4) with else

(9)

(5) where

, , and is the number of where internal nodes in a single wire element [Fig. 1(b)]. This type of basis function offers a compromise between rate of convergence and simplicity of the formulation, with correctly vanishing current at the wire ends. On substituting (4) into (1) and integrating, as [21] the electric field can be expressed in terms of

(6) with (7)

with

and (10) (11) (12) (13) and the reflection and Inversion of yields the sought or in (9) transmission coefficients via (6). The choice of with . is irrelevant, for these factors appear as The only remaining dependence on the sign is in the exponential factor in .

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Fig. 2. (a) Plan view of single cell of dipole-based FSS. We use N = 30 internal nodes, is in the yz plane, j j = 1 V=m, and f = 3 GHz. (b) Shells of points in k -space for evaluation of (9).

E

E

Regarding convergence, in the thin-wire approximation currents are confined to the wire axis but fields are evaluated on the is shifted to for , wire surface [2], i.e., This introduces an exponential decay in (9). Hence, increasing the wire radius accelerates convergence. Depending on the type [5]. For of basis functions, the series may not converge for the triangular basis functions, however, convergence is guaran(see Section IV). This occurs more genteed even when erally for types of basis functions that yield correct vanishing of . These basis funccurrents at the wire ends tions exhibit a decaying polynomial factor that is independent of (see Section VI).

Fig. 3.

Real part of cumulative total impedance for selected matrix elements.

IV. DIPOLE FSS We now study contributions to the overall convergence of (9) for an FSS consisting of a single rectilinear -directed dipole per unit cell [Fig. 2(a)]. Iteration is governed by two parameters and . We consider concentric square shells of “radius” of points in -space [Fig. 2(b)]. For , the “shell” is a , ); for , the shell contains the single point ( eight points (1,0), (1,1), (0,1), ( 1,1), ( 1,0), ( 1 1), (0, 1), , we add increasingly larger (1, ), etc. Starting with shells until convergence1 is reached. Fig. 3 shows the cumulative total impedance for several elements as a function of . Clearly the impedance elements have , because they all still exhibit growing not converged at , which has already converged after oscillations, except for the first step. This follows [20, ch. 4] from (9), because the real part of a self impedance element has only one contribution, i.e., , ). The other elements oscillate, and that by ( is large, i.e., if elements are furmore rapidly so when ther apart (see Fig. 2). for Fig. 4 shows cumulative totals for . Although the curves oscillate at different rates, they all exhibit the same envelope. This is proved in the Appendix. The functional form of the oscillations is derived below. All converge to certain nonzero values. However, the maxare much larger than their asymptotic imum values for values, the latter being of order 0.1 . Physically, shells with represent contributions from plane-wave vectors 1In this paper, we sum over shells which is the safest as well as most complete

calculation. Others perform summation along rows in k -space, thus omitting terms that are often small [22]. This approach converges faster but may fail when the omitted terms become too large.

Fig. 4. Real part of cumulative total impedance matrix elements Z to Z (symbols); Z is shown as an example (line). All curves exhibit the same envelope.

with small tangential components (i.e., small and ) that fluctuate slowly and add more or less constructively. Higher order shells represent larger and that oscillate rapidly and , all slow-oscillating waves interfere destructively. At have been added with relatively few fast-oscillating waves. While several peaks may occur in the convergence envelope (see below), peaks for larger are fortunately usually damped by strongly decaying terms. V. ANALYSIS OF CONVERGENCE We now analyze the various contributions to (9) for in (9) the dipole unit cell. The summands can be expressed analytically, as follows. For -directed

BLACKBURN et al.: NUMERICAL CONVERGENCE IN PERIODIC METHOD OF MOMENTS

dipoles,

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and [where

account of (2)]. From Fig. 2(a), Furthermore

Finally, selecting

on .

, we obtain from (9)

(14) Because of the dipole orientation along , there is only an imin (14) on via . plicit dependence of The and depend linearly on and via (2). We now examine each factor in (14) in turn. 1) The factor is oscillatory because is real regardless of . It oscillates more rapidly for impedance elements corresponding to nodes with large . It depends on the electrical length of the segments . (see Fig. 4) is indepen2) The slow envelope dent of and but depends on . and 3) Since is purely imaginary except for when grating lobes occur, we can write as . This factor decays faster than exponentially, and convergence is accelerated when increasing the electrical radius . 4) Overall convergence is ensured by the monotonic factor , even when . and its components. The Fig. 5 shows Re and monotonic factors cause rapid decay of for increasing . However, their high initial values are suppressed by the -factor. The starts near , overall decay of the envelope of Re after the sinusoid has undergone one cycle and the monotonic factors have decayed to relatively low values. The sinusoidal factor dramatically increases and dominates the time to -cycle in order overall convergence. It takes about one for (9) to converge well. Fig. 6 shows the convergence of for selected other impedance elements. The -factor can be written in terms of by setting for large and not too small , as

We take the -period as defining the required number of steps for convergence. In our example, mm mm whence mm mm . For simequal to those for and plicity,2 we take the limits of sum from 43 to 43, which agrees with the observation from 2Strictly,

this only applies to FSS elements that are symmetric in the plane.

Fig. 5. Plot of Re[S (n = 0; n )] and its individual contributions for varying n only [see (14)]. The monotonic curve (product of third and fourth factors) decays rapidly and has value 600 at n = 2.

Fig. 6. Plot of Re[S shown.

(

n

=0

;n

)] for various matrix element values as

Fig. 4 that

converges well after 50 shells, i.e., . Generalizing these result for other element types, and as-factor is generally the dominant factor in suming that the , if is the larger repeat distance and convergence of each is the smallest segment length, then minimum number of shells for convergence

Thus, the shorter the wire segments relative to the repeat distance, the slower the convergence. The required number of -space points is then per matrix element. In many

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cases, the total length of an FSS element is of the order of . Then, for elements whence

,

time for matrix fill On the other hand, matrix inversion scales as . Thus, -process dominated by matrix filling rather PMM is an than by matrix inversion, which dominates CMM. On a current state-of-the-art desktop computer, CMM is viable up to 1000 elements, whereas PMM is only feasible up to 100 elements per unit cell. Typical calculations use far fewer elements, i.e., between 5 and 30. To illustrate this, a single-layer split ring FSS (analyzed in more detail in Section VII) with 20 nodes requires 10 s/frequency on a Pentium III PC, whereas 40 nodes require . This rate applies only 96 s/frequency, i.e., approaching to elements located in a single plane, for which increasing implies decreasing . For a multilayered FSS, the convergence is faster (i.e., exponential) as a result of . VI. ALTERNATIVE POLYNOMIAL BASIS FUNCTIONS Next we analyze the influence of the choice of basis functions on the convergence rate. For square nonoverlapping pulse functions of length

Fig. 7. Comparison of noncommon factors (19) and (20) in the expressions of S for pulse, triangular, and parabolic basis functions, with s ,s = ,n ,n ,D D : m, l m, and a m.

1 p2 = 1 = 2 2 10

= 30

=

=0 = 0 1 1 = 2 2 10

=

a derivation similar to that in Section III yields

(15)

else an analysis similar to that in Section III yields

(18) It is seen that the polynomial decay factor shows a distinct geometric progression, i.e.

where now, for -directed dipoles (16) , the support of Unlike for nonoverlapping segments; hence the tions. On substituting we obtain

extends only over and are real func-

Fig. 7 shows the frequency dependence of those factors in the , , that are mutually different, i.e. expressions for (19) and

(20) (17) and now result in Compared to triangles, the different -type instead of -type oscillations, and in the factor compared to . Convergence now relies on being sufficiently large. For a fixed number of basis functions, triangles yield greater accuracy. However, of the slow oscillations is the same, since the period the required number of shells remains comparable. Finally, for parabolic basis functions, defined by else

respectively. Note that these decay factors are additional to the , , and . decay imposed by the common factor of , the contributions (19) and (20) are independent of For frequency and take their asymptotic low-frequency (LF) values. on is the same as that Since the linear dependence of on , the quasi-static frequency dependence of (19) and (20) for an arbitrary angle of incidence is similar to variations with the angle of incidence at a fixed frequency. In the LF limit, parabolas yield the smallest values because, (for , triangles would in this example, in the yield the lower values). However, because of numerator for parabolae, both triangles and parabolas show a

BLACKBURN et al.: NUMERICAL CONVERGENCE IN PERIODIC METHOD OF MOMENTS

Fig. 8. Split ring FSS and its reflectivity for normal incidence and z -polarized calculated using thin-wire PMM. Ring radius: 20.5 mm, repeat distance: 49.2 mm, left (right) split opening angle: 30 (20 ) (N = 20). The reflection shows 0.7% rolloff from peak. The result of [23] is shown for comparison, after frequency scaling by 0.9876. This small discrepancy is presumably a result of using straight rather than curved wire segments in our case.

E

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Fig. 9. Convergence of split ring FSS with N = 20; 40; or 60 nodes. Log scale graph (inset) shows later sin oscillations.

-like average decay at high frequencies and are superior to pulses at LFs [5]. Also, (20) oscillates fastest and reaches its first zero at lower frequencies than (19), making parabolas and selecpreferential. Thus, the rate of convergence of depends in part tion of basis functions that minimizes the on the particular frequency range of interest, with parabolas and triangles being superior over one another in specific cases. For . LF problems, parabolas are prefered whenever

VII. MONITORING CONVERGENCE In Section IV, we monitored as a function of shell size in -space, yielding many curves, i.e., one per . However, since the dominant -term is independent of and , it may suffice to analyze a single representative . Thus, we iterate over the (“average”) impedance element in that various , thereby adding the contributions of all shell, yielding a single plot . In practice, the sequence by the latest (i.e., is terminated when the contribution to largest) shell becomes negligibly small. To test this approach, consider the split ring FSS in Fig. 8 [21], [23]. (The figure also validates our model by comparing with published data.) Such an FSS exhibits unusually sharp filtering properties (ultra-high Q), which can be used, e.g., for separating WLAN communication channels [21]. The ring curvature is approximated by straight-wire segments with bend angles at their nodes. We mesh the ring using 20, 40, and 60 nodes and comfor each case in Fig. 9. The peaks pare the convergence of and , respectively. in the slow oscillations occur at Corresponding impedance contributions decrease in magnitude and . Hence, as expected, the to 10% or less when progression is almost linear, but somewhat less perhaps due to

Fig. 10. Convergence of reflection coefficient for split ring FSS (20 nodes; 10, 20, 30, or 40 shells).

the approximated wire curvature. Again, only one cycle of slow oscillations in (14) is needed for convergence. Regarding the convergence of the reflection coefficient itself, Fig. 10 plots its frequency dependence for a split ring FSS with 20 nodes using 10–40 shells. The graph converges with increasing shell numbers, with relatively little change beyond , in accordance with the progression above. The resonance frequency is more sensitive to the mutual interactions than the bandwidth. VIII. CONCLUSION In this paper, we showed that the PMM thin-wire formulation can successfully model wire-type FSSs. Specialized to an

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FSS with unidirectional straight dipoles, the impedance matrix elements show oscillatory behavior with a universal but nonmonotonic envelope as a function of shell radius. Our analysis explains the origin of the oscillations and the convergence rate, including the role of the chosen type of basis function. A geometric progression in the convergence rates of polynomial basis functions has been observed. We described how to monitor convergence of impedance elements by constructing a single represhells are required sentative sequence. It was shown that for each impedance matrix element, yielding an overall com, because the final matrix inversion is putational cost of and is hence not significant.

APPENDIX PROOF OF UNIVERSALITY OF ENVELOPES OF Here we show that all and exhibit the same envelope for all values of and , as shown in Fig. 4. represent linearly modulated complex harmonic The terms -domain functions in the spectral (21) where, provided (22)

. is a real function independent of the “carrier frequency” is an even function of following Moreover, since of (21) is obtained by phase(2), the Hilbert transform shifting over 90 (if ) or 90 (if ), as a result of the product theorem [24]

In words, up to a possible change of sign for its real or is obtained by interchanging imaginary part, the real and imaginary parts of . Therefore, the or Im , generally envelope of either Re defined

by

Re

or

Im , is here simply . This envelope depends neither on the magnitude of nor on its sign. The same result applies to as a rather than , provided , or conversely function of to provided . The same conclusions apply because of the linearity property of the Hilbert to transformation, i.e.

(23)

ACKNOWLEDGMENT The authors thank the reviewers for their comments and suggestions. REFERENCES [1] L. V. Kontorovich and G. P. Akilov, Functional Analysis in Normed Spaces. Oxford, U.K.: Pergamon, 1964. [2] R. F. Harrington, Field Computation by Moment Methods, 1st ed. New York: Macmillan, 1968, ch. 4. [3] R. Mittra, T. Itoh, and T.-S. Li, “Analytical and numerical studies of the relative convergence phenomenon arising in the solution of an integral equation by the moment method,” IEEE Trans. Microwave Theory Tech., vol. 20, pp. 96–104, Feb. 1972. [4] K. J. Webb, P. W. Grounds, and R. Mittra, “Convergence in the spectral domain formulation of waveguide and scattering problems,” IEEE Trans. Antennas Propag., vol. 38, pp. 869–877, Jun. 1990. [5] S. Poulsen, “Scattering from frequency selective surfaces: A continuity condition for entire domain basis functions and an improved set of basis functions for crossed dipoles,” Proc. Inst. Elect. Eng. H Microwave Antennas Propag., vol. 146, no. 6, pp. 234–240, Jun. 1999. [6] A. K. Bhattacharyya, “On the convergence of MoM and mode matching solutions for infinite array and waveguide problems,” IEEE Trans. Antennas Propag., vol. 51, pp. 1599–1606, Jul. 2003. [7] M. Leroy, “On the convergence of numerical results in modal analysis,” IEEE Trans. Antennas Propag., vol. AP-31, pp. 655–659, Jul. 1983. [8] N. Shuley, “A note on relative convergence for moment-method solutions of integral equations of the first kind as applied to dichroic problems,” Electron. Lett., vol. 21, no. 3, pp. 95–97, Jan. 1985. [9] F. S. Johansson, “Convergence phenomenon in the solution of dichroic scattering problems by Galerkin’s method,” Proc. Inst. Elect. Eng. H Microwave Antennas Propag., vol. 134, no. 1, pp. 87–92, Feb. 1987. [10] C. C. Chan and R. Mittra, “On the analysis of frequency-selective surfaces using subdomain basis functions,” IEEE Trans. Antennas Propag., vol. 38, pp. 40–50, Jan. 1990. [11] A. Ishimaru, R. Coe, G. Miller, and P. Geren, “Finite periodic structure approach to large scanning array problems,” IEEE Trans. Antennas Propag., vol. AP-33, pp. 1213–1220, Nov. 1985. [12] A. K. Skrivervik and J. R. Mosig, “Analysis of finite phase arrays of microstrip patches,” IEEE Trans. Antennas Propag., vol. 41, pp. 1105–1114, Aug. 1993. [13] J. M. Usoff and B. A. Munk, “Edge effects of truncated periodic surfaces of thin wire elements,” IEEE Trans. Antennas Propag., vol. 42, pp. 946–953, Jul. 1993. [14] D. Shanks, “Non-linear transformations of divergent and slowly converging sequences,” J. Math. Phys., vol. 34, pp. 1–42, 1955. [15] S. Singh, W. F. Richards, J. R. Zinecker, and D. R. Wilton, “Accelerating the convergence of series representing the free space periodic Green’s function,” IEEE Trans. Antennas Propag., vol. 38, pp. 1958–1962, Dec. 1990. [16] A. Jennings, “Influence of eigenvalue spectrum on the convergence rate of the conjugate gradient method,” J. Inst. Math. Appl., vol. 20, pp. 61–72, 1977. [17] F. X. Canning, “Improved impedance matrix localization method,” IEEE Trans. Antennas Propag., vol. 41, pp. 659–667, May 1993. [18] E. Michielssen and A. Boag, “Multilevel evaluation of electromagnetic fields for the rapid solution of scattering problems,” Microwave Opt. Technol. Lett., vol. 7, pp. 790–795, Dec. 1994. [19] M. G. Floquet, “Sur les équations différentielles linéaires à coefficients périodiques,” Annales d’ École Normale Supérieure, pp. 47–88, 1883. [20] B. A. Munk, Frequency Selective Surfaces: Theory and Design. New York: Wiley, 2000. [21] L. R. Arnaut and J. Blackburn, “High-Q frequency selective surfaces for improved spectral efficiency,” NPL Rep. CETM S155, 2004. [22] L. Henderson, “The scattering of planar arrays of arbitrarily shaped slot and/or wire elements in a stratified dielectric medium,” Ph.D. dissertation, Ohio State Univ., Columbus, 1983. [23] S. Prosvirnin and S. Zouhdi, “Resonances of closed modes in thin arrays of complex particles,” in Advances in Electromagnetics of Complex Media and Metamaterials, S. Zouhdi, Ed. Dordrecht, The Netherlands: Kluwer, 2003, pp. 281–290. [24] E. Bedrosian, “A product theorem for Hilbert transforms,” Proc. IEEE, vol. 51, pp. 868–869, 1963.

BLACKBURN et al.: NUMERICAL CONVERGENCE IN PERIODIC METHOD OF MOMENTS

John Blackburn received the B.Sc. degree in computational physics from Heriot-Watt University, U.K., in 1996 and the Ph.D. degree in superconductivity from Cambridge University, U.K., in 1999. Since then he has been with the National Physical Laboratory, U.K. His work involves theoretical and computational analysis of electromagnetic systems, such as antennas, dielectric resonators, EMC systems, and metamaterials. His work aims to improve measurements of electronic devices and materials important in industry. He has many years of experience writing EM computer simulation codes using such techniques as finite difference, mode matching, and method of moments.

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Luk R. Arnaut (S’94–M’95–SM’01) received the B.S. degree in applied physics and electrical engineering from the University of Gent, Gent, Belgium, in 1989 and the M.Sc. and Ph.D. degrees in communication engineering and digital electronics from the University of Manchester Institute of Science and Technology (UMIST), Manchester, U.K., in 1991 and 1994, respectively. He was a Research Associate with UMIST from 1993 to 1994. During 1994, he was a Postdoctoral Research Scientist with the Defence Research Agency (now QinetiQ), Farnborough, U.K., working on modeling, fabrication, and measurement of synthetic chiral composites. In 1995, he was a Visiting Scientist with the Naval Research Laboratory, Washington, DC. During 1995–1996, he was a Consultant to the Operations Division of British Aerospace (now BAE Systems), Bristol, U.K., responsible for the technical management of the RUSSTECH Programme on hydraulic ram, acoustic lasers (sasers), wave catastrophes, and complex composite materials. Since 1996, he has been a Senior Research Scientist with the U.K. National Physical Laboratory, involved in work on dielectric resonators, structured materials, EMC, antennas, and arrays. He is principal author of more than 60 refereed publications and has received two patents. He is currently co-Convenor of Revision 1 of IEC 61000-4-21 on reverberation chambers for EMC testing (Joint Task Force CISPR(A)/IEC-SC77B). His current research interests include electromagnetic interaction effects, statistical electromagnetics, complex media, and image-processing techniques. Dr. Arnaut is a Fellow of the Institute of Electrical Engineering, U.K. He received the 2003 Rayleigh Prize. He is listed in Who’s Who in Science and Engineering, Who’s Who in the World, and the Cambridge Blue Book.